Tuesday, February 23, 2016

Building Lists of Power Sequences Using Sequence Numbers

A power sequence is a list of the powers of
a given integer.For example the power
sequence for the number 2 consists of the terms 20, 21, 22,
23, 24, etc.These
terms are 1, 2, 4, 8, 16, etc.

But I would like to get a list of many more
terms, and I don’t want to keep multiplying by 2.

What
are Sequence Numbers?

They are integers that have a special
property.When you calculate the decimal
expansion of the inverse of a Sequence Number you get a recognizable number
sequence (many of these sequences are listed in the Online Encyclopedia of
Integer Sequences ( www.OEIS.org )

What kind of mathematics do I need to know
in order to work with Sequence Numbers?

You need to know how to take the inverse of
an integer.The inverse of 123 is “1
over 123” or “1 divided by 123” (1/123).

You also need to know how to do long
division – really long division that you can’t do on your calculator.But don’t worry – you can do it by hand on
paper OR you can get on the internet and go to ( www.wolframalpha.com ) and use this free
“super calculator”.It will take inputs
of about 200 digits, and can provide an output of about 3900 digits.

I would not have been able to do these
calculations without access of the Wolfram Alpha website.And I could not check my answers without the
OEIS website.I would recommend you get
on these websites and play around with them to learn how to use them.

However if you understand how to create the
inverse of an integer, and you understand how to take a fraction and do long
division to get its decimal expansion (how to change a fraction into a
decimal), and you learn how to do
these computations on the internet, then you will have it made.

999,998 is a Sequence Number that will
generate a list of the powers of 2 (2^0, 2^1, 2^2, 2^3, 2^4, etc.), written
in six digit strings.

The Online Encyclopedia of
Integer Sequence does not contain this sequence in their collection.

So how to you build a Sequence Number that
will produce a power sequence, with terms written in strings of any length you
choose?

Well it is easier to do that you might
imagine.

First we start with a 1 and a 0:

10

Then we decide how many digits we want our
terms to be written, and add that many zeros to our number.Suppose we want all of the terms up to 18
digits long.Then after the 10 we attach
000000000000000000 (18 zeros)

10,000,000,000,000,000,000

Next we subtract the number that we want to
calculate the power sequence for.Suppose we choose 5 so that we can generate a list of the powers of 5 (5^0,
5^1, 5^2, 5^3, 5^4, etc.), then we will subtract 5 from the number shown above:

10,000,000,000,000,000,000 – 5 =

9,999,999,999,999,999,995 will be our new
Sequence Number.It will produce a list
of the powers of 5 up to 18 digits long, written in 19 digit strings. I may even produce some accurate 19
digitsSo let’s try it and see if I am
right.

The powers of 5 (5^0, 5^1,
5^2, 5^3, 5^4, etc.)

A Sequence Number that
will generate a list of the powers of 5, written in 19 digit strings is:

One of the things that amazes me about
these numbers is that we did not find many of them until we had the use of
computers.Simply because we could not
perform these operations on a hand calculator, and we were too lazy to do long
division on such large numbers.(I am
included in that bunch not wanting to do the long division by hand.)

The mathematical skills needed to do this
are the ability to find the inverse of a number and to do long division –
really long division.

But since I have access to a computer I can
play with these large numbers and discover their properties.

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About Me

This blog is about a special class of numbers that I call Sequence Numbers. I have been working on them for a few years,and just recently things came together. Phillip is helping me get this material posted.